We present a novel volumetric RPD (restricted power diagram) based framework for approximating the medial axes of 3D CAD shapes adaptively, while preserving topological equivalence, medial features, and geometric convergence. To solve the topology preservation problem, we propose a volumetric RPD based strategy, which discretizes the input volume into sub-regions given a set of medial spheres. With this intermediate structure, we convert the homotopy equivalence between the generated medial mesh and the input 3D shape into a localized problem between each primitive of the medial mesh (vertex, edge, face) and its dual restricted elements (power cell, power face, power edge), by checking their connected components and Euler characteristics. We further proposed a fractional Euler characteristic strategy for efficient GPU-based computation of Euler characteristic for each restricted element on the fly while computing the volumetric RPD. Compared with existing voxel-based or sampling-based methods, our method is the first that can adaptively and directly revise the medial mesh without modifying the dependent structure globally, such as voxel size or sampling density. Compared with the feature preservation method MATFP, our method offers geometrically comparable results with fewer number of spheres, while more robustly captures the topology of the input shape.

翻译：我们提出了一种新颖的基于体积限制幂图（RPD）的框架，用于自适应逼近三维CAD形状的中轴，同时保持拓扑等价性、中轴特征及几何收敛性。为解决拓扑保持问题，我们提出了一种基于体积RPD的策略：给定一组中轴球体，将输入体积离散化为子区域。借助这一中间结构，我们通过检查中轴网格的每个基本元素（顶点、边、面）及其对偶限制元素（幂胞体、幂面、幂边）的连通分量和欧拉示性数，将生成的中轴网格与输入三维形状之间的同伦等价性转化为局部问题。进一步地，我们提出了一种分数阶欧拉示性数策略，以便在计算体积RPD的同时，基于GPU高效地在线计算每个限制元素的欧拉示性数。与现有的基于体素或采样的方法相比，我们的方法是首个能够自适应且直接修正中轴网格而无需全局修改依赖结构（如体素尺寸或采样密度）的方法。与特征保持方法MATFP相比，本方法在使用更少球体数量的情况下，能提供几何上可比的结果，同时更稳健地捕获输入形状的拓扑结构。